Question

Minimum value of $9{\sin }^{2}y+3{\cos }^{2}x+4{\sec }^{2}x+4{\csc }^{2}y$ is

Answer 1

Minimum value of $9si{n}^{2}y+3co{s}^{2}x+4se{c}^{2}x+4cs{c}^{2}y.$

$\Rightarrow (9si{n}^{2}y+4co{s}^{2}y)+3co{s}^{2}x+4se{c}^{2}x$

$\Rightarrow 2\sqrt{9\times 4}+2\sqrt{3\times 4}$

$=2\times 3\times 2+2\times 2\times \sqrt{3}$

$12+4\sqrt{3}$

$=4\sqrt{3}+12$

short trick

$asi{n}^2\theta +bcose{c}^2\theta$

$aco{s}^2\theta +bse{c}^2\theta$

$ata{n}^2\theta +bco{t}^2\theta$

(1) Minimum value

$=2\sqrt{ab}$